In this section, we explain our empirical observation that 1v1 versions of algorithms terminate before their 1vr counterparts with theoretical arguments. In Section REF , we provide a relatively straightforward proof that the 1v1 versions of LUCB and \(\mathcal {A}_1\)  always terminate before their corresponding 1vr versions. In Section REF , we work towards proving that PPR-1v1 always terminates before PPR-1vr, and we reduce this claim to an inequality on beta functions, which we conjecture to be true, and have verified empirically for a range of values. In Section REF , we consider the multi-dimensional version of the PPR martingale stopping rule (denoted PPR-MD), which serves as an alternative for designing a stopping rule for mode estimation. This rule
would use a Dirichlet prior (which is the conjugate of the multinomial distribution), and is analogous to the multi-variate PPR considered in Appendix C of [1]}. We show that PPR-MD always stops after PPR-1v1 on every run.
